1. RADON TRANSFORM A small introduction to RT, its inversion and applications Jaromír Brum Kukal, 2009

2. Johann Karl August Radon Born in Děčín (Austrian monarchy, now North Bohemia, CZ) in 1887 Austrian mathematician living in Vienna Discover the transform and its inversion in 1917 as pure theoretical result No practical applications during his life Died in 1956 in Vienna

3. Actual applications of inverse Radon transform CT – Computer Tomography MRI – Magnetic Resonance Imaging PET – Positron Emission Tomography SPECT – Single Photon Emission Computer Tomography

4. Geometry of 2D Radon transform Input space coordinates x, y Input function f(x, y) Output space coordinates , s Output function F( , s)

5. Theory of pure RT and IRT Radon transform Inverse Radon transform

6. Full circle in RT

7. Shifted full circle in RT

8. Empty circle in RT

9. Shifted empty circle in RT

10. Thin stick in RT

11. Shifted thin stick in RT

12. Full triangle in RT

13. Shifted full triangle in RT

14. Full square in RT

15. Shifted full square in RT

16. Empty square in RT

17. Shifted empty square in RT

18. | x | 2/3 + | y | 2/3 ≤ 1 in RT

19. | x | + | y | ≤ 1 in RT

20. | x | 3/2 + | y | 3/2 ≤ 1 in RT

21. | x | 2 + | y | 2 ≤ 1 in RT

22. | x | 6 + | y | 6 ≤ 1 in RT

23. | x | n + | y | n ≤ 1 for n   in RT

24. 2D Gaussian in RT

25. Shifted 2D Gaussian in RT

26. Six 2D Gaussians in RT

27. Smooth elliptic object in RT

28. Radon transform applications Natural transform as result of measurement: Gamma ray decay from local density map Extinction from local concentration map Total radioactivity from local concentration map Total echo from local nuclei concentration map 3D reality is investigated via 2D slices Artificial realization: Noise – RT – noise – IRT simulations Image decryption as a fun TSR invariant recognition of objects

29. Radon transform properties Image of any f + g is F + G Image of cf is cF for any real c Rotation of f causes translation of F in  Scaling of f in (x,y) causes scaling of F in s Image of a point (2D Dirac function) is sine wave line Image of n points is a set of n sine wave lines Image of a line is a point (2D Dirac function) Image of polygon contour is a point set

30. Radon transform realization Space domain: Pixel splitting into four subpixels 2D interpolation in space domain 1D numeric integration along lines Frequency domain: 2D FFT of original Resampling to polar coordinates 2D interpolation in frequency domain Inverse 2D FFT brings result

31. Inverse transform realization Filtered back projection in space domain: 1D HF filtering of 2D original along s Additional 1D LF filtering along s 2D interpolation in space domain 1D integration along lines brings result Frequency domain: 2D FFT of original Resampling to rectangular coordinates 2D interpolation in frequency domain 2D LF filtering in frequency domain Inverse 2D FFT brings result

32. RT and IRT in Matlab Original as a square matrix D (2 n  2 n ) of nonnegative numbers Vector of angles alpha Basic range alpha = 0:179 Digital range is better alpha = (0:2^N -1)*180/2^N Extended range alpha = 0:359 Output matrix R of nonnegative numbers Angles alpha generates columns of R R = radon(D,alpha); D = iradon(R,alpha); D = iradon(R,alpha,metint,metfil);

33. Reconstruction from 32 angles

34. Reconstruction from 64 angles

35. Reconstruction from 96 angles

36. Reconstruction from 128 angles

37. Reconstruction from 180 angles

38. Reconstruction from 256 angles

39. Reconstruction from 360 angles

40. Reconstruction from 512 angles